To solve for the unknown resistance $R$ in this circuit, we need to analyze the configuration of resistors.

Step 1: Identify the circuit configuration

• The 12 Ω resistor is in series with the combination of the 55 Ω and $R$ resistors, which are in parallel.

Step 2: Determine the total resistance

You mentioned that the total resistance between points A and B is given as 35 Ω. We can use this information to set up an equation.

Parallel combination of 55 Ω and $R$:

The formula for the equivalent resistance $R_{\text{eq}}$ of two resistors in parallel is: $\frac{1}{R_{\text{eq}}} = \frac{1}{R} + \frac{1}{55}$ Solving for $R_{\text{eq}}$: $R_{\text{eq}} = \frac{55R}{55 + R}$

Total resistance from A to B:

The total resistance $R_{\text{total}}$ is the sum of the 12 Ω resistor and the parallel combination: $R_{\text{total}} = 12 + R_{\text{eq}} = 35 \, \Omega$ Substitute the expression for $R_{\text{eq}}$: $12 + \frac{55R}{55 + R} = 35$

Step 3: Solve for $R$

Simplify and solve the equation: $\frac{55R}{55 + R} = 35 - 12$ $\frac{55R}{55 + R} = 23$ Now multiply both sides by $55 + R$: $55R = 23(55 + R)$ $55R = 23 \times 55 + 23R$ $55R = 1265 + 23R$ Subtract $23R$ from both sides: $32R = 1265$ Solve for $R$: $R = \frac{1265}{32} = 39.53 \, \Omega$

So, the value of $R$ is approximately 39.53 Ω.

Would you like more details or further explanation? Here are some related questions for deeper understanding:

1. How do you calculate the total resistance in parallel and series circuits?
2. What are some typical applications of resistors in electrical circuits?
3. How does the value of a parallel resistor affect the total resistance in the circuit?
4. How can Kirchhoff's laws be used to solve complex circuits?
5. What happens to the circuit if the value of $R$ changes significantly?

Tip: Always double-check the configuration of resistors (whether series or parallel) before solving, as it can drastically change the results.